1.
4ˣ+2ˣ⁺¹=80
(2ˣ)² + 2*2ˣ - 80 = 0 ; notezi 2ˣ=t
t²+2*t - 80=0
t²+10t-8t-80=0
t(t+10)-8(t+10)=0
(t+10)*(t-8)=0
t= -10 solutia nu poate fi negativa;
t= 8
2ˣ=8; 2ˣ=2³; x=3
2.
9ˣ-10*3ˣ+9=0;
3²ˣ - 10*3ˣ+9=0
notezi 3ˣ =t
t²-10t+9=0
t²-t-9t+9=0
t(t-1)- 9(t-9)=0
(t-1)*(t-9)=0
t=1
t=9
3ˣ=1; 3ˣ=3⁰ ⇔ x=0
3ˣ= 9; 3ˣ=3² ⇒ x=2
3.
7²ˣ = 6*7ˣ - 5; 7ˣ =t
t² -6*t +5=0
t²-t-5t+5=0
t*(t-1) - 5*(t-1)=0
(t-1)*(t-5)=0
t=1
t=5 (nu este solutie)
7ˣ =1; 7ˣ=7⁰; x=0
4.
10*2ˣ - 4ˣ= 16
16 - 10*2ˣ+ (2ˣ)²=0; notez 2ˣ =t
16-10t+t²=0
16-2t - 8t+t²=0
2*(8-t)- t*(8-t)=0
(8-t)*(2-t)=0
t=8
t=2
2ˣ=8; 2ˣ=2³; x=3
2ˣ=2¹; x=1
5.
36ˣ +7*2ˣ *3ˣ- 78=0
(6ˣ)²+7*6ˣ -78=0; notez 6ˣ =t
t² +7t -78=0
Δ= 49 +312=361
t= (-7 +19)/2= 6
t=(-7-19 )/2= -13; negativ, deci nu poate fi solutie
6ˣ= 6¹; x=1
